Truncation error in series expansions linking microcanonical and canonical ensembles

In the microcanonical thermal pure quantum (mTPQ) method, the canonical ensemble is derived using Taylor series expansions. We prove that the truncation error decreases exponentially with system size when the effective temperature of the mTPQ state is smaller than the target temperature, and otherwise, the error remains constant. We also show the discipline to set the mTPQ parameter by considering the trade-off between the error and the numerical cost.